Letters of administration do the same operational job as letters testamentary — they are the proof of authority institutions demand before releasing anything. The difference is upstream. Letters testamentary follow a will that named someone; letters of administration follow a court's own selection of who should serve.
That selection follows a statutory priority order, generally starting with the surviving spouse and moving through adult children and other relatives. Because nobody was vetted in advance by the person who died, courts commonly require the administrator to post a bond and may impose closer supervision, including approval before major transactions.
You will also see variants. Letters of administration with will annexed, sometimes written c.t.a., issue where a valid will exists but the named executor has died, declined, or been removed — the appointee follows the will's terms while being court-selected. Limited or special letters can issue for a narrow purpose, such as pursuing a lawsuit, without granting full administration authority.
For the family, the practical consequence is delay before anyone has authority at all. Where a will exists, appointment is comparatively routine. Where it does not, there may first be a question of who serves, and until that is resolved nobody can pay a mortgage, cancel a subscription, or stop an account from accruing fees.